We have seen (3.4,2) that the transfer function of a linear/invariant system is a ratio of two polynomials in s, the denominator being the char­acteristic polynomial. The roots of the characteristic equation are the poles of the transfer function, and the roots of the numerator polynomial are its zeros. Whenever a pair of complex poles or zeros lies close to the imaginary axis, a characteristic peak or valley occurs in the amplitude of the frequency – response curve together with a rapid change of phase angle at the corre­sponding value of со. Several examples of this phenomenon are to be seen in the frequency response curves in Figs. 10.3, 10.11, and 10.12. The reason for
this behavior is readily appreciated by putting (3.4,2) in the following form:

G(s) = (д ~ gi) • (a – г2) ~ • • (s – g J (s — Яі) • (« — Я2) • • • (« — Яв)

where the Яг are the characteristic roots (poles) and the zt are the zeros of G(s). Let

(s – Zj.) = рке**

(s – Я*) = гке*^

where p, r, а, (і are the distances and angles shown in Fig. 3.126 for a point s = io) on the imaginary axis. Then

|^| — IT Pfc/П rk

k=1 / fc=l

m n

9 = 2 a* – 2 ft

і i

When the singularity is close to the axis, with imaginary coordinate ш as illustrated for point S on Fig. 3.126, we see that as w passes through a>’, a sharp minimum occurs in p or r, as the case may be, and the angle a or /3 increases rapidly through approximately 180°. Thus we have the following cases:

1. For a pole, in the left half-plane, there results a peak in G and a re­duction in <p of about 180°.

2. For a zero in the left half-plane, there is a valley in |Cr| and an increase in <p of about 180°.

3. For a zero in the right half-plane, there is a valley in |Cr[ and a decrease in cp of about 180°.

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